A summary of the video linked to below: with given constraints on the evaluation points used to generate a RS codeword $\{1, \gamma, \gamma^2, \ldots, \gamma^{q-2} \}$, that codeword of a set of values can be viewed as a set of coefficients to a polynomial, which will have roots $\{ \gamma, \gamma^2, \ldots, \gamma^{n-k} \}$ where $\gamma$ is a generator of the field.
The video leaves the proof as an exercise for the reader, but I’m not sure where to start. I can create test programs to show this is true for specific cases, but I’m not sure where to begin for a mathematical proof. I'm thinking I'd have to choose a set of message values to be encoded $\{ m_0,m_1, ..., m_{k-1} \}$ grind out the summations to produce values or to produce coefficients to show they are equal, for both systematic and non-systematic encoding.
This dual view property would allow for more efficient decoding, such as Sugiyama extended Euclid decoder with time complexity $O((n-k)^2)$, versus Gao extended Euclid decoder with time complexity $O((n)^2)$.
rcgldr found a paper UC Berkeley - dual of an RS code, that is very similar to the video. I don't know why the original question was closed, the video covers the details and would be too long to include in this question, and the dual of an RS code is covered in class rooms, books, or in this case a video. It's also clear that what is being asked for is how to approach creating a proof for dual view.
It would help if a reviewer could explain why this question needs more details or clarity.
A prior question and answer cover this with a different approach:
Dual of generalized Reed-Solomon code
That question and answer are somewhat terse. Expanding on the answer, $f(x)$ is the encoding function evaluated at $\{1, \alpha, \alpha^2, ..., \alpha^{n-1} \}$, while $g(x)$ is a polynomial with defined roots $\{ \alpha, \alpha^2, ..., \alpha^{n-k} \}$
For this approach, for the suggested proof "One then needs to show that the product ($c(x) \cdot c'(x)$) is actually the zero polynomial, using Lagrange interpolation.", where $c(x)$ encoding is based on $f(x)$ and $c'(x)$ encoding is based on $g(x)$. I'm still not sure if this approach is a proof that the dual view of a RS code will have roots $\{ \alpha, \alpha^2, ..., \alpha^{n-k} \}$
There is another question and answer related to this, showing that for systematic encoding, both original and BCH type encoding result in identical encoding matrices, but in this case, the original view is least significant term first, while BCH view is most significant term first, so the roots of the BCH view are reciprocals of $\{ \alpha, \alpha^2, ... \alpha^{n-k} \}$, which are $\{ \alpha^{k}, \alpha^{k+1}, ..., \alpha^{q-2} \}$. Apparently this was a self-discovery by rcgldr.
